Conic intrinsic volumes of Weyl chambers
Volume 9, Issue 3 (2022), pp. 357–375
Pub. online: 7 June 2022
Type: Research Article
Open Access
Open Access
Received
14 January 2022
14 January 2022
Revised
9 May 2022
9 May 2022
Accepted
9 May 2022
9 May 2022
Published
7 June 2022
7 June 2022
Abstract
A new, direct proof of the formulas for the conic intrinsic volumes of the Weyl chambers of types ${A_{n-1}}$, ${B_{n}}$ and ${D_{n}}$ is given. These formulas express the conic intrinsic volumes in terms of the Stirling numbers of the first kind and their B- and D-analogues. The proof involves an explicit determination of the internal and external angles of the faces of the Weyl chambers.
References
Abramson, J., Pitman, J.: Concave majorants of random walks and related Poisson processes. Combin. Probab. Comput. 20(5), 651–682 (2011). MR2825583. https://doi.org/10.1017/S0963548311000307
Abramson, J., Pitman, J., Ross, N., Uribe Bravo, G.: Convex minorants of random walks and Lévy processes. Electron. Commun. Probab. 16, 423–434 (2011). MR2831081. https://doi.org/10.1214/ECP.v16-1648
Alsmeyer, G., Kabluchko, Z., Marynych, A., Vysotsky, V.: How long is the convex minorant of a one-dimensional random walk? Electron. J. Probab. 25, 1–22 (2020). MR4147518. https://doi.org/10.1214/20-ejp497
Amelunxen, D., Bürgisser, P.: Intrinsic volumes of symmetric cones and applications in convex programming. Math. Program. 149(1-2, Ser. A), 105–130 (2015). MR3300458. https://doi.org/10.1007/s10107-013-0740-2
Amelunxen, D., Bürgisser, P.: Probabilistic analysis of the Grassmann condition number. Found. Comput. Math. 15(1), 3–51 (2015). MR3303690. https://doi.org/10.1007/s10208-013-9178-4
Amelunxen, D., Lotz, M., McCoy, M.B., Tropp, J.A.: Living on the edge: phase transitions in convex programs with random data. Information and Inference 3(3), 224–294 (2014). MR3311453. https://doi.org/10.1093/imaiai/iau005
Amelunxen, D., Lotz, M.: Intrinsic volumes of polyhedral cones: A combinatorial perspective. Discrete Comput. Geom. 58(2), 371–409 (2017). MR3679941. https://doi.org/10.1007/s00454-017-9904-9
Bagno, E., Garber, D.: Signed partitions – A balls into urns approach. Preprint at arXiv:1903.02877 (2019). MR4408200
Bagno, E., Biagioli, R., Garber, D.: Some identities involving second kind Stirling numbers of types B and D. Elect. J. Combin. 26(3), 3–9 (2019). MR3982318. https://doi.org/10.37236/8703
Bala, P.: A 3-parameter family of generalized Stirling numbers. Preprint at https://oeis.org/A143395/a143395.pdf (2015)
Bóna, M. (ed.): Handbook of Enumerative Combinatorics. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL (2015). MR3408702
Dowling, T.A.: A class of geometric lattices based on finite groups. J. Combinatorial Theory Ser. B 14, 61–86 (1973). MR0307951. https://doi.org/10.1016/s0095-8956(73)80007-3
Drton, M., Klivans, C.J.: A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010). MR2644900. https://doi.org/10.1090/S0002-9939-10-10369-4
Gao, F.: The mean of a maximum likelihood estimator associated with the Brownian bridge. Electron. Comm. Probab. 8, 1–5 (2003). MR1961284. https://doi.org/10.1214/ECP.v8-1064
Gao, F., Vitale, R.A.: Intrinsic volumes of the Brownian motion body. Discrete Comput. Geom. 26(1), 41–50 (2001). MR1832728. https://doi.org/10.1007/s00454-001-0023-1
Hug, D., Schneider, R.: Random conical tessellations. Discrete Comput. Geom. 56(2), 395–426 (2016). MR3530973. https://doi.org/10.1007/s00454-016-9788-0
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990). https://doi.org/10.1017/CBO9780511623646
Kabluchko, Z., Vysotsky, V., Zaporozhets, D.: Convex hulls of random walks: expected number of faces and face probabilities. Adv. Math. 320, 595–629 (2017). MR3709116. https://doi.org/10.1016/j.aim.2017.09.002
Kabluchko, Z., Vysotsky, V., Zaporozhets, D.: Convex hulls of random walks, hyperplane arrangements, and Weyl chambers. Geom. Funct. Anal. 27(4), 880–918 (2017). MR3678504. https://doi.org/10.1007/s00039-017-0415-x
Klivans, C.J., Swartz, E.: Projection volumes of hyperplane arrangements. Discrete Comput. Geom. 46(3), 417–426 (2011). MR2826960. https://doi.org/10.1007/s00454-011-9363-7
Lang, W.: On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers. Preprint at arXiv:1707.04451 (2017)
Pitman, J.: Combinatorial Stochastic Processes. Lecture Notes in Mathematics. Springer (2006). MR2245368
Schneider, R.: Combinatorial identities for polyhedral cones. St. Petersburg Math. J. 29(1), 209–221 (2018). MR3660692. https://doi.org/10.1090/spmj/1489
Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer (2008). MR2455326. https://doi.org/10.1007/978-3-540-78859-1
Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Canad. J. Math. 6, 274–304 (1954). MR0059914. https://doi.org/10.4153/cjm-1954-028-3
Sloane (editor), N.J.A.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org
Sparre Andersen, E.: On the number of positive sums of random variables. Scandinavian Actuarial Journal 1949(1), 27–36 (1949). MR0032115. https://doi.org/10.1080/03461238.1949.10419756
Sparre Andersen, E.: On the fluctuations of sums of random variables. Math. Scand. 1, 263–285 (1953). MR0058893. https://doi.org/10.7146/math.scand.a-10385
Sparre Andersen, E.: On the fluctuations of sums of random variables II. Math. Scand. 2, 195–223 (1954). MR0068154
Stanley, R.P.: An introduction to hyperplane arrangements. In: Miller, E., Reiner, V., Sturmfels, B. (eds.) Geometric Combinatorics. IAS/Park City Mathematics Series, vol. 13, pp. 389–496. AMS (2007). MR2383131. https://doi.org/10.1090/pcms/013/08
Suter, R.: Two analogues of a classical sequence. J. Integer Seq. 3(1), 1–18 (2000). MR1750742
Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995). MR1311028. https://doi.org/10.1007/978-1-4613-8431-1
